$\mathbb{P}\mathbb{T}$-Symmetric Quantum Liouvillean Dynamics

We discuss a combination of unitary and antiunitary symmetry of quantum Liouvillean dynamics, in the context of open quantum systems, which implies a $D_2$ symmetry of the complex Liouvillean spectrum. For sufficiently weak system-bath coupling, it implies a uniform decay rate for all coherences, i.e., off-diagonal elements of the system’s density matrix taken in the eigenbasis of the Hamiltonian. As an example, we discuss symmetrically boundary driven open $XXZ$ spin $1/2$ chains.

Figure 1

The Liouvillean spectrum $\{{\lambda }_k\}$ for an open $XXZ$ chain with $\Delta =1/2$, $n=4$, maximum driving $\mu =1$, and $\gamma =0.02<{\gamma }_{\mathrm{PT}}$ (top), $\gamma =0.2>{\gamma }_{\mathrm{PT}}$ (middle), and $\gamma =2>{\gamma }_{\mathrm{PT}}$ (bottom). Note the dihedral symmetry of the spectrum with the horizontal and vertical symmetry lines, ${\ell }_{\mathrm{h}}=\mathbb{R}$ and ${\ell }_{\mathrm{v}}=-\gamma +\mathrm{i}\mathbb{R}$, respectively. Because the $XXZ$ chain has a global conservation law $M^{\mathrm{z}}=\sum _j{\sigma }_j^{\mathrm{z}}$, we consider only the most relevant sector with zero total magnetization $M^{\mathrm{z}}=0$.